Let two lattices ${\Lambda }^{\text{'}},{\Lambda }^{\text{'}\text{'}}\subset {ℝ}^s$ have the same number of points on each hyperbolic surface $|x₁...x_s|=C$. We investigate the case when Λ’, Λ” are sublattices of ${\mathbb{Z}}^s$ of the same prime index and show that then Λ’ and Λ” must coincide up to renumbering the coordinate axes and changing their directions.